Tag: Solutions
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Problem 2.13 (Schroeder’s Intro to Thermal Physics)
Problem 2.13 Fun with logarithms.(a) Simplify the expression . (That is, write it in a way that doesn’t involve logarithms.)(b) Assuming that , prove that . (Hint: Factor out the a from the argument of the logarithm, so that you can apply the approximation of part (d) of the previous problem.) Solution: Problem 2.13 Solution…
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Problem 2.12 (Schroeder’s Intro to Thermal Physics)
Problem 2.12 The natural logarithm function, ln, is defined so that for any positive number x.(a) Sketch a graph of the natural logarithm function.(b) Prove the identities and .(c) Prove that (d) Derive the useful approximation ,which is valid when . Use a calculator to check the accuracy of this approximation for x = 0.1…
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Problem 2.10 (Schroeder’s Intro to Thermal Physics)
Problem 2.10 Use a computer to produce a table and graph, like those in this section, for the case where one Einstein solid contains 200 oscillators, the other contains 100 oscillators, and there are 100 units of energy in total. What is the most probable macrostate, and what is its probability? What is the least…
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Problem 2.3 (Schroeder’s Intro to Thermal Physics)
Problem 2.3 Suppose you flip 50 fair coins.(a) How many possible outcomes (microstates) are there?(b) How many ways are there of getting exactly 25 heads and 25 tails?(c) What is the probability of getting exactly 25 heads and 25 tails?(d) What is the probability of getting exactly 30 heads and 20 tails?(e) What is the…
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Problem 2.1 (Schroeder’s Intro to Thermal Physics)
Problem 2.1 Suppose you flip four fair coins.(a) Make a list of all the possible outcomes, as in Table 2.1.(b) Make a list of all the diferent “macrostates” and their probabilities.(c) Compute the multiplicity of each macrostate using the combinatorial formula 2.6, and check that these results agree with what you got by brute force…
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Problem 2.9 – Griffith’s Intro to QM
Problem 2.9 For the wave function in Example 2.2, find the expectation value ofH, at time t=0, the “old fashioned” way: Compare the result we got in Example 2.3. Note: Because is independent of time, there is no loss of generality in using t=0. Solution: Problem 2.9 Solution (Download)
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Problem 1.16 – Griffith’s Intro to QM
Problem 1.16 A particle is represented (at time ) by the wave function (a) Determine the normalization constant A.(b) What is the expectation value of x?(c) What is the expectation value of p? (Note that you cannot get it from . Why not?)(d) Find the expectation value of .(e) Find the expectation value of .(f)…
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Problem 1.14 – Griffith’s Intro to QM
Problem 1.14 Let be the probability of finding the particle in the range , at time t.(a) Show that where What are the units of ? Comment: J is called the probability current, because it tells you the rate at which probability is “flowing” past the point x. If is increasing, then more probability is…
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Problem 3.9 – Griffith’s Intro to QM
Problem 3.9 (a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum.(b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a continuous spectrum.(c) Cite a Hamiltonian from Chapter 2 (other than the finite square well) that has both a discrete and…